6,203 research outputs found
Distributed Approximation of Minimum Routing Cost Trees
We study the NP-hard problem of approximating a Minimum Routing Cost Spanning
Tree in the message passing model with limited bandwidth (CONGEST model). In
this problem one tries to find a spanning tree of a graph over nodes
that minimizes the sum of distances between all pairs of nodes. In the
considered model every node can transmit a different (but short) message to
each of its neighbors in each synchronous round. We provide a randomized
-approximation with runtime for
unweighted graphs. Here, is the diameter of . This improves over both,
the (expected) approximation factor and the runtime
of the best previously known algorithm.
Due to stating our results in a very general way, we also derive an (optimal)
runtime of when considering -approximations as done by the
best previously known algorithm. In addition we derive a deterministic
-approximation
Distributed Connectivity Decomposition
We present time-efficient distributed algorithms for decomposing graphs with
large edge or vertex connectivity into multiple spanning or dominating trees,
respectively. As their primary applications, these decompositions allow us to
achieve information flow with size close to the connectivity by parallelizing
it along the trees. More specifically, our distributed decomposition algorithms
are as follows:
(I) A decomposition of each undirected graph with vertex-connectivity
into (fractionally) vertex-disjoint weighted dominating trees with total weight
, in rounds.
(II) A decomposition of each undirected graph with edge-connectivity
into (fractionally) edge-disjoint weighted spanning trees with total
weight , in
rounds.
We also show round complexity lower bounds of
and
for the above two decompositions,
using techniques of [Das Sarma et al., STOC'11]. Moreover, our
vertex-connectivity decomposition extends to centralized algorithms and
improves the time complexity of [Censor-Hillel et al., SODA'14] from
to near-optimal .
As corollaries, we also get distributed oblivious routing broadcast with
-competitive edge-congestion and -competitive
vertex-congestion. Furthermore, the vertex connectivity decomposition leads to
near-time-optimal -approximation of vertex connectivity: centralized
and distributed . The former moves
toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an
centralized exact algorithm while the latter is the first distributed vertex
connectivity approximation
Sparse Fault-Tolerant BFS Trees
This paper addresses the problem of designing a sparse {\em fault-tolerant}
BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph of the
given network such that subsequent to the failure of a single edge or
vertex, the surviving part of still contains a BFS spanning tree for
(the surviving part of) . Our main results are as follows. We present an
algorithm that for every -vertex graph and source node constructs a
(single edge failure) FT-BFS tree rooted at with O(n \cdot
\min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS
tree rooted at . This result is complemented by a matching lower bound,
showing that there exist -vertex graphs with a source node for which any
edge (or vertex) FT-BFS tree rooted at has edges. We then
consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees}
for short, aiming to provide (following a failure) a BFS tree rooted at each
source for some subset of sources . Again, tight bounds
are provided, showing that there exists a poly-time algorithm that for every
-vertex graph and source set of size constructs a
(single failure) FT-MBFS tree from each source , with
edges, and on the other hand there exist
-vertex graphs with source sets of cardinality , on
which any FT-MBFS tree from has edges.
Finally, we propose an approximation algorithm for constructing
FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result
stating that there exists no approximation algorithm for these
problems under standard complexity assumptions
Effective Edge-Fault-Tolerant Single-Source Spanners via Best (or Good) Swap Edges
Computing \emph{all best swap edges} (ABSE) of a spanning tree of a given
-vertex and -edge undirected and weighted graph means to select, for
each edge of , a corresponding non-tree edge , in such a way that the
tree obtained by replacing with enjoys some optimality criterion (which
is naturally defined according to some objective function originally addressed
by ). Solving efficiently an ABSE problem is by now a classic algorithmic
issue, since it conveys a very successful way of coping with a (transient)
\emph{edge failure} in tree-based communication networks: just replace the
failing edge with its respective swap edge, so as that the connectivity is
promptly reestablished by minimizing the rerouting and set-up costs. In this
paper, we solve the ABSE problem for the case in which is a
\emph{single-source shortest-path tree} of , and our two selected swap
criteria aim to minimize either the \emph{maximum} or the \emph{average
stretch} in the swap tree of all the paths emanating from the source. Having
these criteria in mind, the obtained structures can then be reviewed as
\emph{edge-fault-tolerant single-source spanners}. For them, we propose two
efficient algorithms running in and time, respectively, and we show that the guaranteed (either
maximum or average, respectively) stretch factor is equal to 3, and this is
tight. Moreover, for the maximum stretch, we also propose an almost linear time algorithm computing a set of \emph{good} swap edges,
each of which will guarantee a relative approximation factor on the maximum
stretch of (tight) as opposed to that provided by the corresponding BSE.
Surprisingly, no previous results were known for these two very natural swap
problems.Comment: 15 pages, 4 figures, SIROCCO 201
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